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In this paper we establish some Ostrowski type inequalities for double integral mean of absolutely continuous functions. An application for special means is given as well.

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Let f: R N C be a periodic function with period 2π in each variable. We prove suffcient conditions for the absolute convergence of the multiple Fourier series of f in terms of moduli of continuity, of bounded variation in the sense of Vitali or Hardy and Krause, and of the mixed partial derivative in case f is an absolutely continuous function. Our results extend the classical theorems of Bernstein and Zygmund from single to multiple Fourier series.

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For the fractional dyadic derivative and integral, the following analogues of two theorems of Lebesgue  are proved: the theorem on differentiation of the indefinite Lebesgue integral of  an integrable function at its Lebesgue points, and the theorem on reconstruction of an absolutely  continuous function by means of its derivative. Dyadic fractional analogues of the formula of integration by  parts are also obtained. In addition, some theorems are proved on dyadic fractional differentiation and  integration of a Lebesgue integral depending on a parameter.  Most of the results are new even for dyadic derivatives and integrals of natural order.

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definition of a general fractional integral. Throughout the paper we will suppose that the positive integral operator kernel F : I → (0,∞) defined below is an absolutely continuous function on interval I ⊂ _ _ ℝ. Definition 2.2. Let I be an interval

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